# Properties

 Label 1200.3.bg Level $1200$ Weight $3$ Character orbit 1200.bg Rep. character $\chi_{1200}(193,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $72$ Newform subspaces $17$ Sturm bound $720$ Trace bound $21$

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## Defining parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.bg (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$17$$ Sturm bound: $$720$$ Trace bound: $$21$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(1200, [\chi])$$.

Total New Old
Modular forms 1032 72 960
Cusp forms 888 72 816
Eisenstein series 144 0 144

## Trace form

 $$72q + O(q^{10})$$ $$72q - 24q^{13} + 8q^{17} - 96q^{23} - 128q^{31} - 48q^{33} - 8q^{37} + 64q^{41} + 64q^{43} + 192q^{47} + 192q^{51} + 56q^{53} - 64q^{61} - 480q^{67} - 512q^{71} + 232q^{73} - 648q^{81} + 544q^{83} + 288q^{87} + 576q^{91} - 96q^{93} - 56q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(1200, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1200.3.bg.a $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-24$$ $$q-\beta _{1}q^{3}+(-6+6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.b $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q+\beta _{1}q^{3}+(-4+4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.c $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q-\beta _{1}q^{3}+(-4+4\beta _{2}+3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.d $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q-\beta _{3}q^{3}+(-4+4\beta _{1}-4\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots$$
1200.3.bg.e $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+\beta _{1}q^{3}+(-3+3\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.f $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+\beta _{1}q^{3}+(-2+2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.g $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+5\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots$$
1200.3.bg.h $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+2\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots$$
1200.3.bg.i $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+3\beta _{1}q^{7}-3\beta _{2}q^{9}-6q^{11}+\cdots$$
1200.3.bg.j $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+6\beta _{1}q^{7}-3\beta _{2}q^{9}+18q^{11}+\cdots$$
1200.3.bg.k $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{3}q^{3}+(1+2\beta _{1}+\beta _{2})q^{7}-3\beta _{2}q^{9}+\cdots$$
1200.3.bg.l $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+\beta _{1}q^{3}+(2-2\beta _{2}+\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.m $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+\beta _{1}q^{3}+(4-4\beta _{2}+2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.n $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+\beta _{1}q^{3}+(4-4\beta _{2}-3\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.o $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$20$$ $$q-\beta _{1}q^{3}+(5-5\beta _{2}-2\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.p $$4$$ $$32.698$$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$24$$ $$q-\beta _{1}q^{3}+(6-6\beta _{2}-\beta _{3})q^{7}+3\beta _{2}q^{9}+\cdots$$
1200.3.bg.q $$8$$ $$32.698$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{3}+(\beta _{3}-\beta _{6})q^{7}-3\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(1200, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(1200, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 2}$$